Type I and II Censoring

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Module 15

Saurav De

Department of Statistics Presidency University

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Censoring in particular is a key issue in survival analysis.

Censoring distinguishes survival analysis from regular statistical problems.

Censoring is when an observation is incomplete due to some random cause.

The cause of censoring is usually dependent on the event of interest.

Saurav De (Department of Statistics Presidency University)MLE under Survival Data: Type I and II Censoring 2 / 28

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Type I and II Censoring

Censoring differs from truncation in that the incomplete nature of the observations in truncation occurs due to a systematic selection process inherent to the study design.

Based on the directions through which incompleteness in the observations comes, cencoring is of three types

• Right Censoring • Left Censoring • Interval Censoring

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Right censoring : The most common form of censoring

Here the lifetime of an item is followed until some time at which the event (i.e. failure or death) is yet to occur; but the event takes no further part in the study after that time.

e.g. A lung cancer patient is recruited for clinical trial to test the effect of a drug on his survival from his disease.

But he died in a car accident after T years of his disease.

=⇒ his survival with lung cancer is at least T years, but exact years can not be known.

=⇒ right censored.

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Type I and II Censoring

Left censoring : This occurs when the event of interest has already taken place at the time of observation; but the exact time of occurrance of the event is not known.

e.g.

Onset of an asymptomatic illness, like Brain Cancer

Infection with a sexually transmitted disease like HIV / AIDS

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Interval Censoring :

Here the exact time of the occurrance of the event is not known precisely, but an interval bounding this time is known

In case the interval is too short (e.g. 1 day or 1 hr etc.) the common practice is to ignore the interval censoring and to set one end-point of the interval consistently

e.g.

failure of a machine during Chinese New Year celebration Infection with a sexually transmitted disease like HIV / AIDS in between two annual check-up

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Type I and II Censoring

Depending on how censoring mechanism will work, there are three broad types of censoring

• Type I Censoring • Type II Censoring • Random Censoring We will discuss in brief

above three types in right censoring form

the MLEs of the corresponding parameters under the survivorship probability models

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Type II censoring:

Supposenrandom sample units are set on life-testing experimentation But due to some reasons the experiment terminates after smallestr readings

Let these be denoted by the order statisticsT₍₁₎, . . . ,T_(r₎. Here integerr is prefixed i.e. nonrandom.

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Type I and II Censoring

Since the remaining n−r random sample values are atleast as high as T_(r₎ =⇒ the sampling scheme is a censored one.

Such a censoring is known as Type II censoring.

Type II censoring are frequently used in life-testing experiments.

Here say total of n items are placed on test.

Now instead of continuing until alln items get spared, suppose the experimenter waits just for the first r failures.

Such test saves both time and money.

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Let Ti denote the lifetime / failure time ofith item.

SupposeTi’s be iid having a continuous distribution with pdf f_θ(t) and cdf F_θ(t) whereθ: parameter of the distribution.

Then given t₍₁₎, . . . ,t_(r₎; the realization of T₍₁₎, . . . ,T_(r₎,the likelihood of θ under Type II censoring is

L(θ) = n!

(n−r)!f_θ(t₍₁₎). . .f_θ(t_(r₎)

F_θ(t_(r₎)n−r

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Type I and II Censoring

Verification:

From theory of order statistics, the jount pdf of all the ordeer statistics T₍₁₎, . . . ,T_(n) is

h_θ(t₍₁₎, . . . ,t_(n)) =n!

n

Y

i=1

f_θ(t_(i₎)

=⇒ the marginal joint pdf of T₍₁₎, . . . ,T_(r₎ att₍₁₎, . . . ,t_(r₎ will be gθ(t₍₁₎, . . . ,t_(r₎) =

Z . . .

Z n!

n

Y

i=1

fθ(t_(i))dt_(n). . .dt_(r₊₁₎

=n!

r

Y

i=1

f_θ(t_(i)) Z

. . .







∞

Z

t(n−1)

f_θ(t_(n))dt_(n)





f_θ(t_(n−1)). . . fθ(t_(r₊₁₎)dt(n−1). . .dt_(r+1)

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=n!

r

Y

i=1

f_θ(t_(i)) Z

. . .







∞

Z

t_(n−2)

(1−F_θ(t_(n−1)))f_θ(t_(n−1))dt_(n−1)





f_θ(t_(n−2)). . .

=n!

r

Y

i=1

f_θ(t_(i)) Z

. . .







∞

Z

t(n−3)

(1−Fθ(t(n−2)))²

2 f_θ(t_(n−2))dt_(n−2)





f_θ(t_(n−3)). . .

= n!

2×3

r

Y

i=1

f_θ(t_(i)) Z

. . .







∞

Z

t(n−4)

(1−F_θ(t_(n−3)))³f_θ(t_(n−3))dt_(n−3)







f_θ(t_(n−4)). . .

=⇒ finally we get gθ(t₍₁₎, . . . ,t_(r)) = _(n−r)!^n!

r

Y

i=1

fθ(t_(i))[Fθ(t_(r₎)]^n−r.

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Type I and II Censoring

But given the realizations, the form of joint pdf =⇒ the likelihood of θ.

Hence the form of the likelihood is verified.

Here F_θ(t) : the survival function at the time pointt.

Illustration : Let the of an item be exponential with meanθ.

=⇒ the pdf f_θ(t) = ¹_θexp{−t/θ}

and the survival function at the time point t is F_θ(t) = exp{−t/θ}

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=⇒ under Type II censoring, the likelihood function of θwill be L(θ) = n!

(n−r)!

1 θ^r exp

"

−

r

X

i=1

t_(i)+ (n−r)t_(r)

! /θ

#

=⇒ l(θ) = Const −rlogθ−

r

X

i=1

t_(i)+ (n−r)t_(r₎ θ

=⇒ l⁰(θ) =−r

θ+

r

X

i=1

t_(i)+ (n−r)t_(r) θ²

=⇒ the unique solution of likelihood equation l⁰(θ) = 0 will be

θˆ=

r

X

i=1

t_(i)+ (n−r)t_(r₎ r

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Type I and II Censoring

From SOC we can ensure that ˆθ maximisesL(θ) i.e. ˆθ is the MLE of θ under type II censoring.

Note. θˆ=

r

X

i=1

T_(i₎+(n−r)T_(r₎

r is the MVUE ofθ.

Verification. Joint pdf ofT₍₁₎, . . . ,T_(r) is

g_θ(t_(·)) = n!

(n−r)!

1 θ^r exp











−

r

X

i=1

t_(i)+ (n−r)t_(r₎ θ









 DefineZ₁ =nT₍₁₎ , Z_i = (n−i+ 1)(T_(i₎−T_(i₋₁₎) ;i = 2, . . . ,r

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Check that the Jacobian of transformation from (T₍₁₎, . . . ,T_(r₎)−→(Z1, . . . ,Zr) is ^(n−r_n!^)! and

r

X

i=1

Z_i =

r

X

i=1

T_(i₎+ (n−r)T_(r₎=rθ.ˆ

=⇒ the joint pdf of Z₁, . . . ,Z_r is

h_θ(z) = 1 θ^r exp











−

r

X

i=1

zi

θ











which implies that Z₁, . . . ,Z_r are iid exponential (mean =θ) random variables.

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Type I and II Censoring

=⇒ E_θ(ˆθ) =E_θ

r

X

i=1

Z_i

!

/r =θ.

Also g_θ(t_(·))∈ an OPEF. =⇒ the statistic ˆθ=

r

X

i=1

T_(i₎+(n−r)T_(r₎

r is

complete sufficient. Hence by Lehman-Scheffe Theorem the Notefollows.

Type I Censoring

Sometimes experiments are run over a fixed period of time 3the exact lifetime of an item will be known only if it is less than some

pre-determined value.

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In such a situation data are said to be type I censored (from right).

More precisely a type I censored sample is one that arises when n items numbered say 1,2, . . . ,n are subject to limited periods of observations, and

let L₁, . . . ,L_n be those periods3

ith item’s lifetimeT_i is observable only if T_i ≤L_i. L_i : called fixed censoring time for ith item

If allL_i are equal, data are said to be single type I censored.

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Type I and II Censoring

Assume that T_is are iid with common pdff_θ(t) and survival function F_θ(t).

Fromith item we record the exact the exact lifetime Ti as the realization provided Ti ≤Li.OtherwiseLi is recorded as the realization.

Let Y_i denote the potential response (the response which is surely obtained) from ith item.

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Then

Y_i = T_i if T_i ≤L_i (called uncensored case)

= Li ifTi >Li (called censored case) for all i. =⇒ Y_i = min{T_i,L_i}.

Also define indicator variables

δi = 1 ifTi ≤Li (called uncensored case)

= 0 ifT_i >L_i (called censored case) Then δ_is are called censoring indicators.

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Type I and II Censoring

So the type I censored data can be represented by the pairs of random variables (Y_i, δ_i) for all i.

=⇒ the jount likelihood of θfor given data set{(t_i, δi),i = 1, . . . ,n} on (Y_i, δ_i)s will be

L(θ) =

n

Y

i=1

[f_θ(t_i)]^δⁱ

F_θ(L_i)(1−δ_i)

How this is obtained ? It is true thatP_θ[Yi =yi |δi = 0] = 1 ifyi =Li. P_θ[Y_i =y_i, δ_i = 0] = P_θ[δ_i = 0] =P_θ[T_i >L_i] ify_i =L_i

= F_θ(L_i) i.e. the likelihood for the ith item is

Li(θ) =Fθ(Li) ifδi = 0(⇔ yi =Li) . . . (∗)

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Again

P_θ[Yi ≤yi, δi = 1] = P_θ[Ti ≤yi] (asδi = 1⇔ Ti ≤Li ⇔ Yi =Ti)

= Fθ(yi)

=⇒ L_i(θ) =f_θ(y_i) ifδ_i = 1 . . . (∗∗) (∗) and (∗∗) =⇒ Li(θ) = [fθ(ti)]^δⁱ

Fθ(Li)(1−δ_i)

As pairs (Y_i, δ_i)s are independent, the joint likelihood of θ will be L(θ) =

n

Y

i=1

[f_θ(t_i)]^δⁱ

F_θ(L_i)(1−δ_i)

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Type I and II Censoring

Suppose the readings (in some suitable unit) of life from 10 items, set on an experimentation, are as follows:

1.4^∗ , 0.17 , 1.4^∗ , 1.4^∗ , 0.28 , 0.94 , 1.4^∗ , 0.7 , 1.07 , 1.20

where reading with ^∗ is censored from right. If the life distribution is Weibull with density

f(t) =αβt^β−1e^−αt^β,t >0 ; α, β >0,

and also α= 1, find the ML estimate of β from the life data readings.

Computation. From the nature of censoring, the data are type I censored from right and has the common censoring time point 1.4.

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Also here β is the only unknown parameter to be estimated.

Note that for the given Weibull distribution, F(t) =e^−αt^β.Hence the likelihood function of β (withα= 1) will be

L(β) =

10

Y

i=1

{f(ti)}^δⁱ

F(L) ^1−δⁱ

where δi : censoring indicator and L: the common censoring time (= 1.4 here). Therefore

L(β) =β^r

10

Y

i=1

t_i^(β−1)δⁱe

− 10

X

i=1 δ_it_i^β

e^−(10−r)L^β

wherer =

10

X

i=1

δ_i = number of uncensored cases. t_is denote exact readings.

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Type I and II Censoring

So the loglikelihood of β will be

`(β) =rlnβ+ (β−1)

10

X

i=1

δilnti−

10

X

i=1

δit_i^β−(10−r)L^β

So ∂

∂β`(β) = r

β +

10

X

i=1

δ_ilnt_i −

10

X

i=1

δ_it_i^βlnt_i −(10−r)L^βlnL.

Hence the likelihood equation ofβ reduces to the form

β=r

"₁₀ X

i=1

δ_it_i^βlnt_i+ (10−r)L^βlnL−

10

X

i=1

δ_ilnt_i

#⁻¹

. . . (∗)

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(∗) does not have any explicit solution. So we have to solve it numerically (using Newton Raphson method) for the ML estimate of β. To find out the initial value of β, we used the quantile method.

R program for the solution of numerical equation (∗) : R Code and Output :

> t = c ( 1 . 4 , 0 . 1 7 , 1 . 4 , 1 . 4 , 0 . 2 8 , 0 . 9 4 , 1 . 4 , 0 . 7 , 1 . 0 7 , 1 . 2 0 )

> del = c (0 ,1 ,0 ,0 ,1 ,1 ,0 ,1 ,1 ,1)

> max . i t e r = 1 0 0

> r = sum ( del )

> # i n i t i a l v a l u e of ’ beta ’

> qu = q u a n t i l e ( t )

> i n i t = log ( log (4) ) / log ( as . n u m e r i c ( qu [ 4 ] ) )

> i n i t

[1] 0 . 9 7 0 7 6 1 4

> b e t a = N U L L

> b e t a [ 1 ] = i n i t

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Type I and II Censoring

R Code and Output (continued) :

> # 1 st d e r i v a t i v e f u n c t i o n

> f u n 1 = f u n c t i o n ( b ) {

s u m 1 =0 s u m 2 =0

for ( i in 1: l e n g t h ( t ) ) {

s u m 1 = s u m 1 +( del [ i ] * log ( t [ i ]) ) }

for ( j in 1: l e n g t h ( t ) ) {

s u m 2 = s u m 2 +( del [ j ] * (( t [ j ]) ^ b ) * log ( t [ j ]) ) }

l1 =( r / b ) +( s u m 1 ) - sum2 -((10 - r ) * ( ( 1 . 4 ) ^ b ) * log ( 1 . 4 ) ) r e t u r n ( l1 )

}

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R Code and Output (continued) :

> # 2 nd d e r i v a t i v e f u n c t i o n

> f u n 2 = f u n c t i o n ( b ) {

s u m 1 =0

for ( i in 1: l e n g t h ( t ) ) {

s u m 1 = s u m 1 +( del [ i ] * (( t [ i ]) ^ b ) * log ( t [ i ]) * log ( t [ i ]) ) }

l2 = -( r / ( b * b ) ) - sum1 -((10 - r ) * ( ( 1 . 4 ) ^ b ) * log ( 1 . 4 ) * log ( 1 . 4 ) ) r e t u r n ( l2 )

}

> for ( k in 2: max . i t e r ) {

b e t a [ k ]= b e t a [ k -1] -( f u n 1 ( b e t a [ k - 1 ] ) / f u n 2 ( b e t a [ k - 1 ] ) ) if ( b e t a [ k ] - b e t a [ k - 1 ] < 0 . 0 0 0 0 0 0 1 )

b r e a k }

> # MLE of ‘ beta ’ ( c o n v e r g e d v a l u e )

> b e t a [ k ] [1] 1 . 2 4 4 8 8 7